Clay Mathematics Monographs 2 Volume 2 Lecture Notes on Motivic on Notes Lecture

The notion of a motive is an elusive one, like its namesake “the motif” of Cezanne’s impressionist method of painting. Its existence was fi rst suggested by Grothendieck in 1964 as the underlying structure behind the myriad cohomology theories in Algebraic Geometry. We now know that there is a triangulated theory of motives, discovered by , which suffi ces for the development Lecture Notes on of a satisfactory Motivic Cohomology theory. However, the existence of motives themselves remains conjectural. This book provides an account of the triangulated theory of motives. Its purpose Motivic Cohomology is to introduce Motivic Cohomology, to develop its main properties, and fi nally to relate it to other known invariants of algebraic varieties and rings such as Milnor K-theory, étale cohomology, and Chow groups. The book is divided into lectures, grouped in six parts. The fi rst part presents the defi nition of Motivic Cohomology, based upon the notion of presheaves with transfers. Some elementary comparison theorems are given in this part. The theory of (étale, Nisnevich, and Zariski) sheaves with transfers is developed in parts two, three, and six, respectively. The theoretical core of the book is the fourth part, presenting the triangulated category of motives. Finally, the comparison with higher Chow groups is developed in part fi ve. The lecture notes format is designed for the book to be read by an advanced graduate student or an expert in a related fi eld. The lectures roughly correspond to one-hour lectures given by Voevodsky during the course he gave at the Institute for Advanced Study in Princeton on this subject in 1999–2000. In addition, many of Carlo Mazza

the original proofs have been simplifi ed and improved so that this book will also Vladimir Voevodsky be a useful tool for research mathematicians. Charles Weibel Mazza, Voevodsky and WeibelVoevodskyand Mazza,

For additional information and updates on this book, visit www.ams.org/bookpages/cmim-2

www.ams.org American Mathematical Society CMIM/2 AMS www.claymath.org CMI Clay Mathematics Institute

4 color process 232 pages pages on 50 lb stock • 1 3/16 spine